Boundary state and output feedbacks for underactuated systems of coupled time-fractional PDEs with different space-dependent diffusivity

Abstract

This paper considers the boundary stabilisation problem of an underactuated system of coupled time-fractional partial differential equations (PDEs) with different space-dependent diffusivity by state and observer-based output feedback. For this underactuated system, there are n fractional equations, and n−1 of n equations are actuated at the boundary inputs, i.e. n−1 available inputs control the entire system of n fractional equations. In other words, we do not need to control all equations. Using the backstepping mapping, the Dirichlet boundary controller is derived to enable the Mittag–Leffler stability of the closed-loop system with the aid of the Lyapunov direct method. The Mittag–Leffler convergence of designed boundary observer and output feedback stabilisation are subsequently proved. Finally, a fractional numerical example is provided to support the effectiveness of the proposed synthesis for the case when the kernel matrix PDE has not an explicit solution.

Keywords:

• Fractional PDEs
• underactuated systems
• Mittag–Leffler stability
• space-dependent diffusivity
• backstepping

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Estonian Research Council [grant number PRG658], the COST Action CA15225, a network supported by COST (European Cooperation in Science and Technology), and National Natural Science Foundation of China [grant number 61807016].

Notes on contributors

Juan Chen

Juan Chen received her Ph.D. degree in control science and engineering from school of IoT Engineering and master degree in applied mathematics from school of science, Jiangnan University, China, in 2018 and 2011. She joined the Mechatronics, Embedded Systems and Automation (MESA) Lab of the University of California, Merced, USA as an exchange Ph.D. student hosted by Prof. YangQuan Chen from March 2015 to September 2015. She was a postdoctoral researcher at Tallinn University of Technology, Estonia, from December 2018 to December 2019. She is currently working as an assistant professor at Changzhou University, China and a Member of IEEE. Her research interests mainly focus on fractional-order systems and control, and control of diffusion systems.

Aleksei Tepljakov

Aleksei Tepljakov received his Ph.D. in Information and Communication Technology from Tallinn University of Technology in 2015. From January 2016 Dr. Tepljakov holds a Research Scientist position at the Department of Computer Systems, School of Information Technologies, Tallinn University of Technology. His main research interests include fractional-order modelling and control of complex systems and developing efficient mathematical and 3D modelling methods for virtual and augmented reality for educational and industrial applications. From March 2017 he serves as head of Re:creation Virtual Reality laboratory in the Centre for Intelligent Systems. Dr. Tepljakov is a Senior Member of IEEE with 8 years of service, a member of IEEE Control Systems Society since 2012, and a member of Education Society from 2018.

Eduard Petlenkov

Eduard Petlenkov was born in 1979. He received his B.Sc, M.Sc and PhD degrees in computer and systems engineering from Tallinn University of Technology. He is a Professor in the Department of Computer Systems at Tallinn University of Technology. His main research interests lie in the domain of nonlinear control, system analysis and computational intelligence.

Bo Zhuang

Bo Zhuang received his Ph.D. degree from school of IoT Engineering of Jiangnan University, China. He joined the MESA Lab of the University of California, Merced, USA as an exchange Ph.D. student hosted by Prof. YangQuan Chen from December 2015 to June 2016. He is now an associate professor in Binzhou University, Shangdong, China. His research interests include analysis and control of distributed parameter systems, mobile agents, and coverage optimisation.